Simple Harmonic Motion exists whenever an object is being pulled towards an equilibrium point by a force that is proportional to the displacement from the equilibrium point. Two common examples of SHM are masses on a spring (one that obeys Hooke’s Law) and a pendulum (with a small angle displacement)
👨💻 Enduring Understanding 3.B
Classically, the acceleration of an object interacting with other objects can be predicted using F = ma.
🏘 Essential Knowledge 3.B.3
Restoring forces can result in oscillatory motion. When a linear restoring force is exerted on an object displaced from an equilibrium position, the object will undergo a special type of motion called simple harmonic motion
** Note - for AP 1, we can assume that all the springs used are ideal springs. If you plan on taking the AP C: Mech exam, that will not be the case
Periods of a Pendulum and Spring
Amplitude is the height of the motion measured from the equilibrium point.
Period is the time that it takes for an object to complete one full cycle of its motion. The period is measured in seconds and is the inverse of the frequency (measured in Hz).
In the above equation, 𝜔 is the angular velocity of the object. This will be covered in detail in Unit 7: Torque & Rotational Motion
For a pendulum, the period of the oscillation can be described using the equation:
Where L is the length of the pendulum, and g is the acceleration due to gravity.
Looking at the equation, we can see that the period is proportional to the square root of the length. So a shorter pendulum will have a shorter period, and vice versa. In fact in order to double the period, we’d need to quadruple the length.
The period for a mass on a spring has a very similar equation. The only main difference is that the spring’s period doesn’t depend on length and acceleration due to gravity, but rather the mass hung on the spring and the spring constant. For a more in-depth derivation, check out this link
Key Features of Pendulums & Springs
When dealing with pendulums and springs, a lot of the questions you’ll be dealing with refer to the velocities, forces, and accelerations at various locations in the oscillation:
The net force and acceleration vectors always point in the same direction.
The force and acceleration vectors are greatest when the spring is fully stretched and compressed
The velocity vector is 0 at the extremes where the spring is fully stretched and compressed.
The velocity vector is at its maximum when the mass passes through the equilibrium point. (This is also where the force and acceleration vectors are 0)